---
tag: S8RG6N1PRT
kind: concept
name: Laplace's equation
msc: [35J05, 31B05]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace%27s_equation
same_as:
  wikidata: Q339444
  wikipedia: Laplace's equation
---

## Summary

Laplace's equation is the [[partial differential equation|GC5BDZ6QBC]] $\Delta u = 0$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]], the sum of the second partial derivatives.
Its solutions are the [[harmonic functions|0DEB0JMN0W]].
It is the model elliptic equation.
It describes steady states: equilibrium temperatures, electrostatic and gravitational potentials in empty space, and potential flows of an ideal fluid.
The [[Dirichlet problem|8D56GQ4AG8]] asks for a solution with given values on the boundary of a region.

## Explanations

### L1 human

Laplace's equation says that at every point the value of $u$ equals the average of its values nearby.
A stretched elastic membrane fixed along a wire frame takes a shape whose height satisfies it.
So does the temperature in a metal plate once it has stopped changing.
In the plane, $u(x, y) = x^2 - y^2$ is a solution, because $u_{xx} = 2$ and $u_{yy} = -2$ add up to zero.